The continuum limit of the Kuramoto model on sparse random graphs

In this paper, we study convergence of coupled dynamical systems on convergent sequences of graphs to a continuum limit. We show that the solutions of the initial value problem for the dynamical system on a convergent graph sequence tend to that for the nonlocal diffusion equation on a unit interval, as the graph size tends to infinity. We improve our earlier results in [Arch. Ration. Mech. Anal., 21 (2014), pp. 781--803] and extend them to a larger class of graphs, which includes directed and undirected, sparse and dense, random and deterministic graphs. There are three main ingredients of our approach. First, we employ a flexible framework for incorporating random graphs into the models of interacting dynamical systems, which fits seamlessly with the derivation of the continuum limit. Next, we prove the averaging principle for approximating a dynamical system on a random graph by its deterministic (averaged) counterpart. The proof covers systems on sparse graphs and yields almost sure convergence on time intervals of order $\log n,$ where $n$ is the number of vertices. Finally, a Galerkin scheme is developed to show convergence of the averaged model to the continuum limit. The analysis of this paper covers the Kuramoto model of coupled phase oscillators on a variety of graphs including sparse Erd\H{o}s-R{\' e}nyi, small-world, and power law graphs.